# Observational Evidence of the Dark matter Existence

### Problem 1

Analyze the problem of a small satellite galaxy moving in the gravitational field of a large galaxy (the problem of non-decreasing behavior of the rotational curves.)

### Problem 2

What radial dependence of the density of a spherically symmetric galactic halo corresponds to the constant velocity of satellite galaxies?

### Problem 3

We have noted above that outside the optically observed range of a galactic disk the rotation curves of the dwarf satellite galaxies do not depend on the galacto-centric distance (the flatness of the rotational curves). External regions of the galaxies are mostly filled by cold neutral hydrogen. Evidently the rotation velocity of the gas gives important information about the mass distribution in the galaxy. How can one measure that velocity?

### Problem 4

Using the virial theorem, express the mass of a galaxy cluster through the observed quantities---the average velocity of galaxies in the cluster and its size. Estimate the mass of the Coma cluster (Berenice's hair) for $R\approx10^{23}m$, $\langle v^2\rangle^{1/2}\approx2\cdot10^6m/s$.

### Problem 5

Show that the age of the matter--dominated Universe contradicts observations.

### Problem 6

A galaxy has the visible mass of $10^{11}M_\odot$ and a horizontal rotation curve up to distance of $30kpc$ at velocity $250km/s$. What is the ration What is the dark to visible mass ratio in the galaxy?